derivative of log_{10}((arcsin(x)^2))

解答

\frac{d}{d x} (lo g_{10} ((arcsin (x))^{2}))

\frac{2}{ln ( 10 ) arcsin ( x ) - x ^{2} + 1}

求解步骤

\frac{d}{d x} (lo g_{10} ((arcsin (x))^{2}))

使用对数计算法则:

lo g_{a} (b) = \frac{ln ( b )}{ln ( a )}

= \frac{d}{d x} \frac{ln ( ( arcsin ( x ) ) ^{2} )}{ln ( 10 )}

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

= \frac{1}{ln ( 10 )} \frac{d}{d x} (ln ((arcsin (x))^{2}))

使用连锁律:

\frac{1}{( arcsin ( x ) ) ^{2}} \frac{d}{d x} ((arcsin (x))^{2})

= \frac{1}{( arcsin ( x ) ) ^{2}} \frac{d}{d x} ((arcsin (x))^{2})

\frac{d}{d x} ((arcsin (x))^{2}) = \frac{2 arcsin ( x )}{1 - x ^{2}}

= \frac{1}{ln ( 10 )} \cdot \frac{1}{( arcsin ( x ) ) ^{2}} \cdot \frac{2 arcsin ( x )}{1 - x ^{2}}

化简

\frac{1}{ln ( 10 )} \cdot \frac{1}{arcsin ^{2} ( x )} \cdot \frac{2 arcsin ( x )}{1 - x ^{2}} : \frac{2}{ln ( 10 ) arcsin ( x ) - x ^{2} + 1}

= \frac{2}{ln ( 10 ) arcsin ( x ) - x ^{2} + 1}

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